Oliver Smart

Oliver Smart

(c) O.S. Smart 1995, all rights reserved

*This section is adapted from part 1.2.2. of O.S. Smart,
"Simulation of a conformational rearrangement of the substrate in
D-xylose Isomerase", University of London Ph.D. thesis (1991). *

Subject to the Born-Oppenheimer approximation the potential energy for a molecule of

where **R_i** are the position vectors of the nuclei and the
**Z_i** the charges. The electronic wavefunction and potential energy
are given by:

where **r_i** are the position vectors for the electrons.

In the case of the simplest molecule the exact solution
can be found numerically, but with some difficulty
(Wind, H., 1965, *J. Chem. Phys.* **42**:2371-2373).
In more complicated systems, it is necessary to take a different
approach. The LCAO method, which is commonly used, expresses a
trial molecular orbital (MO) as a Linear Combination of the
individual Atomic Orbitals
for the molecule:

The **variation principle** can be exploited to find values for the
coefficients . The principle states that the expectation value
. of a
Hamiltonian
calculated with any function
:

must lie above the true ground state energy (the integration is over all space). Or put another way if we have a trial wavefunction for the molecule and we make a change which lowers the energy then the new function is a better approximation to the true wavefunction to the system. Thus, the optimal value for the coefficients can be found by adjusting the coefficients so as to minimize .

A further complication arises because of the last term of the
Hamiltonian which describes electron-electron repulsion.
Without this term it would be possible to use the
method of separation of variables
to solve the Schrödinger equation independently for each
electron, but this is not possible when the term is taken into
consideration. The standard approximation, developed by Hartree
and Fock, to avoid this problem is to assume that each electron
moves in the average field due to the nuclei and the other
electrons.
An initial set of wavefunctions is guessed
for the molecule. In the LCAO approach a series of fixed **phi**
called the basis set, is chosen (this must at least give a
representation for each occupied atomic orbital of each atom)
and the initial guess is a set of coefficients **C_k** (n.b.
this must satisfy the Pauli principle and be antisymmetric - see
Hinchliffe).
The optimal
coefficient for each atomic orbital in turn is found while keeping
the other orbitals frozen. The procedure is continued until all
the coefficients remain unaltered in a pass through the cycle and
become self-consistent. The method is known as the self-consistent
field approach (SCF).

This is commonly known as *i.e.,* varying **R_i** so as to minimize
the potential energy). Its great problem lies in its c.p.u. cost: which rises
by approximately **n** to the power 3.5 where **n**
is the number of basis
functions used for the calculation.
Therefore, it is very expensive to improve the
accuracy of a simulation for a small molecule, or to model a larger
molecule. The practical limit of the calculation is a few tens of
atoms
(Weiner *et al.*, 1984; Singh and Kollman, 1984).
Semi-empirical quantum
simulations (such as AM1: Dewar *et al.* , 1985), in which
approximations and empirical data are introduced into SCF calculations allow this
limit to be extended. However the techniques are far too expensive
to be applied to the smallest protein.

If you want to learn more about quantum chemistry then look at this bibliography .

or the advanced topic:
**The application of quantum chemistry to protein simulation.**
*(also contains references for this section)*.